Instead of the plane wavefronts of Fraunhofer, we now consider spherical wavefronts.
i.e. we use Huygens' principle from a point source at a finite distance.
We must also consider any possible obliquity factor. The more analytic formulation of the Huygens-Fresnel theory developed by Kirchhoff provides an expression for this factor:
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When q = p, K(q) = 0 so there is no back travelling wave.
When q = p/2, K(q) = 1/2, get half amplitude for wavelets perpendicular to the primary wavefront. See figure 35a.
Consider the propagation of wavefronts from S to P. Using Huygens' principle the disturbance at P is due to wavelets generated over a spherical wavefront passing through O, a distance r from S. This is illustrated in figure 35b.
As the angle f increases the path difference for the wavelets seen at P increases. If R = rsinf then:
D = D1+D2 = r-Ö{r2-R2}+r-Ö{r2-R2}
providing R << r, i.e. f small.
We can use the binomial expansion to express the square roots as a series of terms and ignore high order terms if f is small:
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We can divide the spherical wavefront up into annula zones such that the path difference D changes by l/2 as we pass from the inner to the outer radius of each annulus. These are called half period zones. They are illustrated in figure 36.
Using the path difference approximation above:
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where Rm is the outer radius of the mth zone.
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The area of each zone (for f small) is given by:
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which is independent of Rm (or m) within the approximation.
The wavelets constructed by Huygens' principle from within a single zone are in phase to within ±p when they reach P. However the mean phase of wavelets from the zones changes by p as we move from zone to zone.
The total amplitude seen at P from all the wavelets from all half period zones is given by a complex sum:
A = A1-A2+A3-¼(-1)m-1Am¼
The sign alternates because of the phase change from 1 zone to the next.
The A's slowly vary because:
The first 2 cancel and the obliquity factor dominates at large m so the A's decrease monotonically.
We can group the terms in 2 ways to find the limiting sum of all the zones.
First consider the sum of an odd number of terms grouped in 2 ways:
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All the terms in brackets () are just positive because the A's decrease monotonically hence:
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But A1 » A2 and Am » Am-1 so:
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Similarly we can consider the sum of an even number of terms, this gives:
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So as m® large we must have:
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This is a rather neat result. The amplitude at P due to all the wavelets from the spherical wavefronts from S is half the amplitude from the first half period zone.
In the above we lumped together all the wavelets from each zone. In fact we can divide each zone up into n sub-zones of equal area. As we traverse each zone the phase changes by p/n between each sub-zone. Therefore the phasor diagram for the summation, sometimes called a vibration curve, is a spiral with each zone approximating to a semicircle as shown in figure 37. The sum asymptotes to the centre of the spiral by the above argument.
Suppose we introduce an aperture which blocks some of the half period zones. This will create a zone plate as illustrated in figure 38. Phasor diagrams to show what happens are given in figure 39.
In particular consider a circular aperture which is the same size as the first half period zone. This removes all but the first semicircle of the vibration curve spiral. The amplitude we get at P is now A1. This is twice the amplitude we had without the aperture. Thus the intensity is 4 times the unobstructed value. By blocking the light we have increased the intensity at P!
Conversely suppose we block the first zone with a small disc. The vibration curve remaining gives an amplitude of » A1/2. We see an intensity of A1A1*/4 where there should be a shadow.
Poisson, an ardent opponent of Fresnel's wave theory, predicted the latter consequence using the theory and cited it as damning proof that it must be wrong. He was shown to be mistaken when Arago demonstrated that there was indeed a bright spot at the centre of a shadow of a small disc. Theorists beware.
We can go a stage further and construct a zone plate which consists of a set of concentric annuli that block alternate half period zones. The amplitude sum for the disturbance as P is now:
A = A1+A3+A5+A7+¼An or A = A2+A4+A6+A8+¼An
where n is the index of the outer open zone. Whether we start with the first zone blocked or open the result is approximately the same. We get a very large amplitude at P.
We have not considered what amplitude we get if we move P laterally away from the axis defined by S and the centre of the plate. The new position P¢ defines its own set of half period zones which overlap the zone plate annuli off-axis. Even at small displacements each P¢ zone is crossed by a large number of the plate zones so the resulting amplitude is very small. As P¢ is moved the fraction of P¢ odd and even zones covered changes and so the amplitude oscillates up and down.
The Fresnel diffraction pattern of a zone plate consists of a very bright central spot surrounded by very faint rings. The width of the central peak is » (Rn-Rn-1)(r+r)/r. The zone plate is acting exactly like a lens.
Rearranging the formula we derived for the radii of the half period zones:
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The zone plate has a primary focal length fp = R12/l and produces an image by the simple Gaussian lens formula.
Note that the focal length depends directly on the wavenumber. The behaviour as a lens is strongly chromatic.
We also find bright spots at positions given by subsidary focal lengths fp/3, fp/5, fp/7 etc.. Convince yourself that this is the case! Are these spots stronger or weaker than the primary focus?
Suppose we want fp = 100mm operating in light l = 4000Å then the radius of the first zone is 2mm. Even at optical wavelengths zone plates are small.
Zone plates can be constructed to act as the objective lens in a soft X-ray microscope.
The operation of a zone plate is an example of Fresnel or near field diffraction.
To analyse near field diffraction through rectangular apertures we must abandon half period zones.
Consider the path difference between the axial route SOP and the path through point (y,z) on S, SAP shown on figure 40.
Using the Pythagorean theorem on triangles SOA and POA we get:
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Expanding the square roots using the binomial series and neglecting high order terms in y/ro, z/ro, y/ro and y/ro gives:
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Because we are in the near field region we now include quadratic terms in y and z although we still approximate to small angles.
We now make a substitution letting:
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So we get:
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Note this has a quadratic dependence on the position within S (the aperture) compared to the linear dependence on the diffraction angle used in the Fraunhofer case.
We integrate to find the resultant amplitude at P:
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Where:
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Once again we assumed K(q) » 1, small angles. u1,u2,v1,v2 define the extent of the aperture.
The integrals are known as the Fresnel Integrals.
Consider the component integral:
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The real and imaginary parts, R(w) and I(w) are well studied, tabulated functions. We can express our resultant amplitude in terms of R(w) and I(w):
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We can expand the complex exponential in the integral definition of R and I into a real and imaginary part giving:
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Therefore dl2 = dR2+dI2 = dw2 so if we plot R(w) and I(w) as functions of w on an argand diagram we get a curve and distances along the curve (dl) are identical to scaled distances across the aperture (dw). Such a plot is called a Cornu Spiral after the french professor Marie Alfred Cornu (1841-1902) who devised it. See figure 41.
w® ¥ the curve ® 0.5+i0.5
w® - ¥ the curve ® -0.5-i0.5
The gradient of the curve:
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So the gradient is zero at w = 0,Ö2,2,¼ and infinite at w = 1,Ö3,Ö5,¼.
We can use the Cornu spiral to calculate Fresnel diffraction patterns for rectangular apertures. This requires that we move P about. This makes the analysis complicated. It is much easier to move the aperture about with respect to the centre line. This is equivalent to moving P in the small angle limit.
The simplest aperture is a single straight edge. For this case u1® ¥,u2® ¥,v2® ¥ and v1 determines the position of the edge wrt the centre line SOP. See figure 42. The corresponding vibration curve (phasor diagram) is shown in figure 43 and the resulting shadow is shown in figure 44.
If v1® -¥ P is in the open well away from the edge. The amplitude is given by 1+i so the intensity is 2.
If v1 = 0 then P is at the position of the geometric shadow and the amplitude is 0.5+i0.5 so the intensity is 0.5 which is 1/4 the unobstructed intensity.
As v1® +ve the amplitude decrease monotonically within the geometric shadow.
As v1® -ve the amplitude oscillates up and down. The first maximum occurs at w = 1.2 followed by a minimum at w = 1.8 etc..
For example let ro = ro = 0.25m. If l = 5000Å then
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So the distance across the screen = w/(5×103). The first peak is at 1.2×0.2 = 0.24mm and the first minimum is at 1.8×0.2 = 0.36mm.
Again we only need to consider the Fresnel integral in one direction.
The difference v2-v1 = Dw is a constant proportional to the slit width. Furthermore Dw represents a constant length along the Cornu spiral. As P move so the position of Dw changes on the spiral. The amplitude at P is proportional to the distance between the ends of the length Dw wound along the spiral.
As the slit is moved wrt the centre line (or equivalently as P is moved off-axis) so the length Dw moves along the spiral. When P is a long way off-axis Dw is tightly wrapped around one or other end of the spiral so the amplitude is small. As P moves towards the axis Dw unwinds and the amplitude grows.
We can plot the amplitude as a function of the mean position of Dw, i.e. (v1+v2)/2. The detailed shape depends on the width of the slit (magnitude of Dw).
If Dw < 3 then we get a single peak. If Dw > 3 we get 2 or more peaks.
Try and work out how you get an approximation to Young's slits interference fringes from the Cornu spiral.
So far we have treated light as a wave phenomenon - amplitude and phase. These can be handled mathematically as a complex quantity.
However we know from EM theory that light is a transverse wave. Therefore we should consider 2 disturbances in orthogonal directions perpendicular to the wavevector.
Consider 2 harmonic components of the same frequency:
Ex = [^x]Eoxcos(wt-kz)
Ey = [^y]Eoycos(wt-kz+f)
where f is a phase difference and [^x] and [^y] are unit vectors. To find the resultant we add the 2 components vectorially. We will consider 3 cases.
(1) f = 0 or 2pn where n is integer.
We get:
E = (Eox[^x]+Eoy[^y])cos(wt-kz)
This is a plane polarized wave of amplitude Ö{Eox2+Eoy2} inclined to the x-axis by q = tan-1(Eoy/Eox).
This is called the P-state or linear light.
(2) f = 2pn+p/2
Ex = [^x]Eocos(wt-kz)
Ey = [^y]Eosin(wt-kz)
E = Eo[cos(wt-kz)[^x]+sin(wt-kz)[^y]]
Now the amplitude is constant since sin2q+cos2q = 1 for all q. The angle the E vector makes with the x-axis is:
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If we fix z then vector rotates at frequency w/(2p).
If we fix t and change z the end of E traces out a right handed helix with a complete rotation in z = l = 2p/k.
This is called the Â-state or right handed circular light.
Similarly if f = 2pn-p/2 we get:
E = Eo[cos(wt-kz)[^x]-sin(wt-kz)[^y]]
This is called the L-state or left handed circular light.
(3) If f is fixed but arbitary and Eox ¹ Eoy then:
Ex = [^x]Eoxcos(wt-kz)
Ey = [^y]Eoycos(wt-kz+f)
The Ex equation gives us that:
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But the Ey equation can be written:
Ey = Eoy[^y][cos(wt-kz)cosf-sin(wt-kz)sinf]
Substituting for cos(wt-kz) and sin(wt-kz) gives:
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This is a standard equation for an ellipse. The amplitude vector still rotates at angular frequency w but the amplitude varies so that E traces an ellipse.
This is called the E-state or elliptical light.
In general:
Â+L = E
but if Eox = Eoy then
Â+L = P
Plane, circular and elliptical light are illustrated in figure 45a,b,c.
(4) If f varies randomly with time then we get the unpolarized state or natural light.
Each atom emits a well defined state lasting » 10-8 seconds (coherence length 3m but there is no correlation between atoms so there is no average of prefered polarization state.